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Prior-expansion score conflict diagnostic

Algorithms AIMech:PriorExpansionScoreConflictDiagnostic DRAFT

A Bayesian prior-data conflict diagnostic that embeds a baseline prior density g(theta) into a smoothly parameterized expansion family g(theta|gamma) with g(theta|gamma_0)=g(theta), evaluates the marginal-likelihood score S(y)=d/d_gamma log p(y|gamma)|_{gamma=gamma_0} via Fisher's identity as the baseline posterior expectation of the log-prior derivative when parameter support is fixed, and calibrates the score discrepancy against prior-predictive or hierarchical mixed-predictive reference distributions.

Identity Notes & Concept Scope No normalized identity match for the prior-expansion score conflict diagnostic across all 442 pinned AIO records and 561 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches including ignored and hidden files across the worktree and repository checkouts returned 0 matches for 1902.10393, 20-BA1204, PriorExpansionScoreConflictDiagnostic and Fisher's identity, and 5 lines for prior expansion (4 locator lines for Gaussian-prior algebraic expansion in GaussianLossSmoothedMixedModelVariationalInference and 1 pending frontier entry for Nott et al. 2020/2021 in 20261013T013528Z-inla-group-node-conflict.json). Assessed 61 semantic AIO rows and 215 native names/definitions, verifying all 61 AIO rows and 214 prior native rows plus reading the INLA group-node conflict diagnostic record. Ten full AIO nodes with incident edges and generated records and nine native conflict-diagnostic and variational context records were verified. This procedure is defined by embedding an elicited prior into a smooth family g(theta|gamma), evaluating S(y)=d/d_gamma log p(y|gamma)|_{gamma=gamma_0} via Fisher's identity at the baseline posterior, and calibrating against prior or mixed predictive draws. It is distinct from unit-level cross-validated prior-likelihood splitting, DAG separator node-splitting, INLA group-node conflict checking, multiple-testing node-conflict maxT calibration, Gaussian variational message conflict screening, and sequential two-stage single-draw randomized score checks.

Claims & Grounded Evidence

7 Claims

Mechanism #prior-expansion-and-fisher-identity

Embed the elicited baseline prior density g(theta) into a smoothly parameterized family of proper priors g(theta|gamma) satisfying g(theta|gamma_0)=g(theta), and define the marginal likelihood p(y|gamma)=integral g(theta|gamma)*p(y|theta) d_theta. The local conflict statistic is the scalar score S(y)=d/d_gamma log p(y|gamma)|_{gamma=gamma_0} (with multi-parameter expansions varied one component at a time). Whenever differentiation with respect to gamma and integration over theta can be interchanged on a gamma-invariant support, Fisher's identity yields S(y)=integral (d/d_gamma log g(theta|gamma)|_{gamma=gamma_0})*g(theta|y) d_theta, which is a posterior expectation under the baseline model gamma=gamma_0 and depends on y only through any minimal sufficient statistic, invariant to its choice.

supports Source: Using prior expansions for prior-data conflict checking · Locator: Section 1, Equations 1.1–1.3, PDF pp. 2–3; Section 2.1–2.2, PDF pp. 4–6
supports Source: Detecting Conflicts in Evidence Synthesis Models Using Score Discrepancies · Locator: Section 2.1, Equation 1, PDF pp. 4–5

Mechanism #divergence-and-relative-belief-sensitivity

Let RB(theta|y,gamma)=g(theta|y,gamma)/g(theta|gamma)=p(y|theta)/p(y|gamma) denote the relative belief ratio. Because p(y|theta) does not depend on gamma, differentiating log RB(theta|y,gamma) with respect to gamma at gamma=gamma_0 gives d/d_gamma log RB(theta|y,gamma)|_{gamma=gamma_0}=-d/d_gamma log p(y|gamma)|_{gamma=gamma_0}=-S(y) for every theta with p(y|theta)>0 (Equation 2.4). Consequently, for the fixed-baseline-weighted relative-belief functional F_alpha(gamma)=(alpha-1)^-1*log integral RB(theta|y,gamma)^(alpha-1)*g(theta|y) d_theta motivated by the Renyi divergence in Equation 2.2 (which is distinct for alpha!=1 from standard fixed-posterior Renyi divergence R_alpha(g(theta|y)||g(theta|gamma))), as well as its alpha->1 limit integral log RB(theta|y,gamma)*g(theta|y) d_theta and the fixed-posterior Kullback–Leibler divergence KL(g(theta|y)||g(theta|gamma)), the derivative with respect to gamma at gamma=gamma_0 equals -S(y) across all orders alpha>0 for which the finite positive likelihood moment E_{g(theta|y)}[p(y|theta)^(alpha-1)] and required log-likelihood/score moments exist and differentiation/limit interchange is justified (Section 2.2, Equation 2.4 and unnumbered KL derivative, PDF pp. 5–6). As an independently derived clarification of the Section 2.2 notation (not an expression explicitly supplied by Nott et al., 2020, PDF pp. 5–6), if the posterior weighting distribution g(theta|y,gamma) inside R_alpha(g(theta|y,gamma)||g(theta|gamma)) is also varied with gamma under those regularity and finite-moment conditions, the derivative at gamma=gamma_0 for finite alpha!=1 adds the order-dependent escort covariance term Cov_{g(theta|y)}(p(y|theta)^(alpha-1), d/d_gamma log g(theta|gamma)|_{gamma=gamma_0})/((alpha-1)*E_{g(theta|y)}[p(y|theta)^(alpha-1)]), which reduces in the Kullback–Leibler limit alpha->1 to the posterior covariance Cov_{g(theta|y)}(log p(y|theta), d/d_gamma log g(theta|gamma)|_{gamma=gamma_0}).

supports Source: Using prior expansions for prior-data conflict checking · Locator: Section 2.1–2.2, Equations 2.2 and 2.4, PDF pp. 5–6

Mechanism #prior-predictive-and-multivariate-calibration

Throughout the prior-expansion score framework, the expansion parameter gamma is taken to be a scalar, or when a baseline prior is embedded into a family with multiple expansion parameters they are varied one by one to construct separate directional checks sensitive to distinct aspects of the prior (such as location shift versus scale or tail heaviness). For each scalar expansion parameter gamma where conflict corresponds to larger values of S(y), calibrate the observed score S(y_obs) against its baseline prior-predictive reference distribution Y~p(y)=p(y|gamma_0) via the upper-tail probability p_S=P(S(Y)>=S(y_obs)), or use the lower tail or a two-sided tail when departures in the opposite or either direction of gamma indicate conflict.

supports Source: Using prior expansions for prior-data conflict checking · Locator: Section 1, Equations 1.1–1.2, PDF pp. 2–3; Section 2.1, PDF pp. 4–5; Section 2.3, PDF p. 7; Section 4, Example 4.3, PDF p. 13; Section 5.2, PDF pp. 19–21, 27

Mechanism #conjugate-exponential-family-equivalence

In conjugate exponential-family settings where g(theta|gamma) perturbs natural hyperparameters, S(y) reduces to differences between posterior expectations of natural sufficient statistics and their prior expectations. In the normal location model with X_1,...,X_n~N(theta,sigma^2) (known sigma^2>0), sufficiency reduction bar_x~N(theta,sigma^2/n), and multiplicative variance-expanded prior theta|gamma~N(mu_0,gamma*tau_0^2) at baseline gamma_0=1, marginal differentiation of log N(bar_x|mu_0,gamma*tau_0^2+sigma^2/n) at gamma=1 and Fisher's identity E_{theta|bar_x}[-1/2+(theta-mu_0)^2/(2*tau_0^2)]=-1/2+(v_post+(m_post-mu_0)^2)/(2*tau_0^2) both yield S_gamma(bar_x)=-tau_0^2/(2*(tau_0^2+sigma^2/n))+tau_0^2*(bar_x-mu_0)^2/(2*(tau_0^2+sigma^2/n)^2)=-1/2+((bar_x-mu_0)^2/(2*tau_0^2))*(tau_0^2/(tau_0^2+sigma^2/n))^2+(sigma^2/n)/(2*(tau_0^2+sigma^2/n)), whereas differentiating directly with respect to the prior variance parameter tau^2 at tau^2=tau_0^2 as in Example 4.1 gives S_{tau^2}(bar_x)=S_gamma(bar_x)/tau_0^2=-1/(2*(tau_0^2+sigma^2/n))+(bar_x-mu_0)^2/(2*(tau_0^2+sigma^2/n)^2). For fixed (tau_0^2,sigma^2/n), both S_gamma(bar_x) and S_{tau^2}(bar_x) are strictly increasing affine functions of (bar_x-mu_0)^2, so the upper-tail score p-value P(S(Y)>=S(bar_x_obs)) under Y~N(mu_0,tau_0^2+sigma^2/n) equals the exact Evans–Moshonov prior-predictive tail P((Y-mu_0)^2>=(bar_x_obs-mu_0)^2)=2*(1-Phi(|bar_x_obs-mu_0|/sqrt(tau_0^2+sigma^2/n))).

supports Source: Using prior expansions for prior-data conflict checking · Locator: Section 2.1, Equation 2.1, PDF p. 4; Section 4, Example 4.1, PDF pp. 10–11

Mechanism #hierarchical-mixed-predictive-extension

For a two-stage hierarchical prior g(theta_1)*g(theta_2|theta_1) expanded as g(theta_1|gamma^(2))*g(theta_2|theta_1,gamma^(1)) around baseline (gamma_0^(1),gamma_0^(2)), define the conditional lower-level score S^(1)(y,theta_1)=integral (d/d_gamma^(1) log g(theta_2|theta_1,gamma^(1))|_{gamma^(1)=gamma_0^(1)})*g(theta_2|theta_1,y) d_theta_2 and eliminate theta_1 by averaging over the observed-data posterior g(theta_1|y_obs) to obtain S^(1)(y)=E[S^(1)(y,theta_1)|y_obs]=integral (d/d_gamma^(1) log g(theta_2|theta_1,gamma^(1))|_{gamma^(1)=gamma_0^(1)})*g(theta_2|theta_1,y)*g(theta_1|y_obs) d_theta. Calibrate S^(1)(y_obs) via p_{S1}=P(S^(1)(Y)>=S^(1)(y_obs)) under the mixed-predictive distribution Y~m(y)=integral g(theta_2|theta_1)*g(theta_1|y_obs)*p(y|theta) d_theta, and calibrate the upper-level score S^(2)(y)=d/d_gamma^(2) log p(y|gamma^(2))|_{gamma^(2)=gamma_0^(2)}=integral (d/d_gamma^(2) log g(theta_1|gamma^(2))|_{gamma^(2)=gamma_0^(2)})*g(theta_1|y) d_theta_1 via p_{S2}=P(S^(2)(Y)>=S^(2)(y_obs)) under the marginal prior-predictive law Y~p(y).

supports Source: Using prior expansions for prior-data conflict checking · Locator: Section 3, Equations 3.1–3.2, PDF pp. 8–10; Section 4, Example 4.3, PDF pp. 12–13

Limitation #support-dependent-expansion-boundary

Fisher's identity in Equation 1.3 requires the support of g(theta|gamma) to be independent of gamma so that differentiation with respect to gamma can be moved inside the integral over theta. When the prior support varies with gamma—as in the symmetrically distorted trine quantum state estimation experiment where the Bloch-disk constraint depends on the distortion parameter gamma—Equation 1.3 is invalid directly; evaluating S(y)=d/d_gamma log p(y|gamma)|_{gamma=gamma_0} requires first reparameterizing the prior and likelihood onto a fixed domain (such as polar coordinates on [0,1]x[0,2*pi]) so that gamma enters the transformed integrand smoothly.

supports Source: Using prior expansions for prior-data conflict checking · Locator: Section 5.2, Equations 5.6–5.8 and polar reparameterization, PDF pp. 23–27

Limitation #reference-simulation-and-approximation-scope

Although Fisher's identity avoids refitting models at gamma!=gamma_0 for the observed dataset y_obs, computing the prior-predictive or mixed-predictive reference distribution of S(Y) when no closed-form sufficient-statistic reduction exists requires re-estimating the baseline posterior g(theta|Y) for each simulated replicate dataset Y. To reduce this cost in non-conjugate models, the source substitutes plug-in point estimates (such as group sample means or least-squares/Moore–Penrose estimates in the LASSO shrinkage check) or suggests variational approximations, which define distinct approximate diagnostics whose tail probabilities are not guaranteed to match exact full-posterior score calibration.

supports Source: Using prior expansions for prior-data conflict checking · Locator: Section 5.1, Examples 5.1–5.2, Equations 5.1–5.3, PDF pp. 14–16; Section 6 Discussion, PDF p. 28
supports Source: Detecting Conflicts in Evidence Synthesis Models Using Score Discrepancies · Locator: Section 2.1, Equation 1 and computational cost discussion, PDF pp. 4–5

⚠️ Curated Limitations & Negative Results

Fisher's identity in Equation 1.3 requires fixed prior support with respect to the expansion parameter gamma so differentiation and integration commute; gamma-dependent support boundaries require reparameterization onto a fixed domain before differentiating (nott2020, Section 5.2, PDF pp. 25, 27).
Exact prior-predictive or mixed-predictive calibration of S(Y) requires evaluating the posterior expectation of d/d_gamma log g(theta|gamma)|_{gamma=gamma_0} across many simulated replicate datasets Y when no closed-form sufficient-statistic reduction exists. Plug-in point-estimate substitutions (such as sample means or least-squares/Moore–Penrose estimates in Section 5.1) and variational approximations alter the statistic and are not identical to full-posterior score calibration.
The score S(y) measures local first-order sensitivity along a scalar expansion direction g(theta|gamma) at gamma=gamma_0; different expansion directions probe different aspects of prior specification (for example, location shift versus tail heaviness or variance inflation), and when multiple expansion parameters are introduced they are varied one by one rather than pooled into a joint multi-parameter test statistic (nott2020, Section 1, PDF pp. 2–3; Section 5.2, PDF pp. 19–21).
In the source manuscript, Section 3 (PDF p. 9, line 24) writes "a p-value for the check of g(theta_2)" when defining p_{S2} for the upper-level prior g(theta_1), and Section 5.1 Example 5.1 (PDF p. 14) prints the summation upper limit in the kurtosis denominator as x instead of n. The curated claims follow the mathematical definitions of the upper-level expansion g(theta_1|gamma^(2)) and the n-group sample kurtosis.
All 31 pages of Nott et al. (2020, arXiv:1902.10393v2) and 52 pages of Yang, Nott, & Presanis (2026, arXiv:2511.02977v2) were read. No author MCMC code, bowhead whale simulation or quantum state tomography dataset was executed; constructed numerical examples in research/discovery/prior-expansion-score-conflict-boundaries.py verify Fisher's identity across non-unit prior and observation variances, fixed-posterior and varying-posterior KL divergence derivatives together with moving-posterior finite-alpha Renyi escort covariance derivatives, conjugate normal equivalence, coordinate-wise one-at-a-time directional checks, hierarchical mixed-predictive score calibration, and support-dependent failure without reparameterization.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:BayesianNetwork` Bayesian Network	related_match	The diagnostic evaluates prior-data conflict in Bayesian models and decomposes multi-stage hierarchical priors g(theta_1)*g(theta_2\|theta_1) along directed Bayesian graphical structures. BayesianNetwork provides model-representation context rather than the prior-expansion score statistic or Fisher-identity predictive calibration; this is a related mapping, not equivalence or subclass membership.
`AIO:RandomEffectsModel` Random Effects Model	related_match	Hierarchical random-effects models are a primary setting for the two-stage mixed-predictive and marginal prior-predictive score checks, but RandomEffectsModel denotes a model class rather than the score-based conflict diagnostic procedure.

Evidence Sources & Primary Literature

2 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`nott2020`	Using prior expansions for prior-data conflict checking David J. Nott, Max Seah, Luai Al-Labadi, Michael Evans, Hui Khoon Ng, Berthold-Georg Englert	primary paper full_text	ID: https://doi.org/10.1214/20-BA1204 sha256:2d36e3af352a…
`yang2026`	Detecting Conflicts in Evidence Synthesis Models Using Score Discrepancies Fuming Yang, David J. Nott, Anne M. Presanis	primary paper full_text	ID: https://doi.org/10.48550/arXiv.2511.02977 sha256:4829e0d63d56…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 4 Argon

Created: 2026-10-17T08:23:00Z

Updated: 2026-10-24T10:55:00Z

Source File: data/curated/PriorExpansionScoreConflictDiagnostic.yaml · Schema: curated_concept.yaml

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